Geometry and Representation Theory of Symplectic Resolutions

代数几何是研究多项式方程组的解集的学科。辛簇是具有长度概念的簇，因此在每个点周围，簇看起来像是一个偶数维向量空间。 辛结构自然出现在经典力学中，经典力学是描述宏观物体运动的物理学分支。 这个研究项目的主要目标是使用线性代数和几何的工具，以更好地理解和利用辛品种所享有的自然对称性，称为辛解析。这项工作有望加强该领域与几何表示论、组合学和弦理论的联系。更详细地说，这个研究项目包括四个相互关联的子项目。第一个子项目是研究辛分解对之间的辛对偶。这种对偶性与三维规范理论的镜像对偶性密切相关。第二个子项目是证明一个关于辛锥的交上同调与其辛分解的量子上同调的猜想。第三个子项目是使用环带镶嵌法对超环面品种进行分类，这与环面品种通过扇形进行分类的方式大致相同。第四个子项目是研究拟阵的一个新的不变量，称为其Kazhdan-Lusztig多项式。

Algebraic geometry is the study of solution sets of systems of polynomial equations called varieties. Symplectic varieties are varieties equipped with a notion of length so that around each point the variety looks like an even-dimensional vector space. Symplectic structures arise naturally in classical mechanics, the branch of physics that describes the motion of macroscopic objects. The main goals of this research project are to use tools from linear algebra and geometry to better understand and exploit natural symmetries enjoyed by symplectic varieties known as symplectic resolutions. The work is expected to strengthen connections of the field with geometric representation theory, combinatorics, and string theory.In more detail, this research project consists of four interrelated subprojects. The first subproject is to study symplectic duality between pairs of symplectic resolutions. This duality is closely related to mirror duality for 3-dimensional gauge theories. The second subproject is to prove a conjecture relating the intersection cohomology of a symplectic cone to the quantum cohomology of its symplectic resolution. The third subproject is to classify hypertoric varieties using zonotopal tilings, much in the same way that toric varieties are classified by fans. The fourth subproject is to study a new invariant of a matroid called its Kazhdan-Lusztig polynomial.